Abstract: Publication date: 1 November 2009
Source:Physica D: Nonlinear Phenomena, Volume 238, Issue 21
Author(s): J. Daunizeau , K.J. Friston , S.J. Kiebel
In this paper, we describe a general variational Bayesian approach for approximate inference on nonlinear stochastic dynamic models. This scheme extends established approximate inference on hidden-states to cover: (i) nonlinear evolution and observation functions, (ii) unknown parameters and (precision) hyperparameters and (iii) model comparison and prediction under uncertainty. Model identification or inversion entails the estimation of the marginal likelihood or evidence of a model. This difficult integration problem can be finessed by optimising a free-energy bound on the evidence using results from variational calculus. This yields a deterministic update scheme that optimises an approximation to the posterior density on the unknown model variables. We derive such a variational Bayesian scheme in the context of nonlinear stochastic dynamic hierarchical models, for both model identification and time-series prediction. The computational complexity of the scheme is comparable to that of an extended Kalman filter, which is critical when inverting high dimensional models or long time-series. Using Monte-Carlo simulations, we assess the estimation efficiency of this variational Bayesian approach using three stochastic variants of chaotic dynamic systems. We also demonstrate the model comparison capabilities of the method, its self-consistency and its predictive power.

Abstract: Publication date: 15 March 2011
Source:Physica D: Nonlinear Phenomena, Volume 240, Issue 7
Author(s): Clayton Bjorland , César J. Niche
Infinite energy solutions to the Navier–Stokes equations in R 2 may be constructed by decomposing the initial data into a finite energy piece and an infinite energy piece, which are then treated separately. We prove that the finite energy part of such solutions is bounded for all time and decays algebraically in time when the same can be said of heat energy starting from the same data. As a consequence, we describe the asymptotic behavior of the infinite energy solutions. Specifically, we consider the solutions of Gallagher and Planchon (2002) [2] as well as solutions constructed from a “radial energy decomposition”. Our proof uses the Fourier Splitting technique of M.E. Schonbek.
Highlights ► We study long-time behaviour of infinite energy solutions to a 2D Navier–Stokes equation. ► The finite energy part of these solutions satisfies a Navier–Stokes like equation. ► Asymptotic behaviour of these is that of the heat equation with the same initial data. ► This, in turn, determines the decay of infinite energy solutions towards specific solutions.

Abstract: Publication date: 15 October 2012
Source:Physica D: Nonlinear Phenomena, Volume 241, Issue 20
Author(s): Dimitrios Giannakis , Andrew J. Majda , Illia Horenko
Many problems in complex dynamical systems involve metastable regimes despite nearly Gaussian statistics with underlying dynamics that is very different from the more familiar flows of molecular dynamics. There is significant theoretical and applied interest in developing systematic coarse-grained descriptions of the dynamics, as well as assessing their skill for both short- and long-range prediction. Clustering algorithms, combined with finite-state processes for the regime transitions, are a natural way to build such models objectively from data generated by either the true model or an imperfect model. The main theme of this paper is the development of new practical criteria to assess the predictability of regimes and the predictive skill of such coarse-grained approximations through empirical information theory in stationary and periodically-forced environments. These criteria are tested on instructive idealized stochastic models utilizing K -means clustering in conjunction with running-average smoothing of the training and initial data for forecasts. A perspective on these clustering algorithms is explored here with independent interest, where improvement in the information content of finite-state partitions of phase space is a natural outcome of low-pass filtering through running averages. In applications with time-periodic equilibrium statistics, recently developed finite-element, bounded-variation algorithms for nonstationary autoregressive models are shown to substantially improve predictive skill beyond standard autoregressive models.
Highlights ► Relative entropy as a metric of predictive skill and model error. ► Long-range predictive skill revealed through coarse-grained partitions of phase space. ► Equilibrium consistency condition in long-range forecasts. ► Predictive fidelity of nonstationary autoregressive models with periodic external factors.

Abstract: Publication date: 15 October 2012
Source:Physica D: Nonlinear Phenomena, Volume 241, Issue 20
Author(s): Franz Achleitner , Peter Szmolyan
Traveling wave solutions of viscous conservation laws, that are associated to Lax shocks of the inviscid equation, have generically a transversal viscous profile. In the case of a non-transversal viscous profile we show by using Melnikov theory that a parametrized perturbation of the profile equation leads generically to a saddle–node bifurcation of these solutions. An example of this bifurcation in the context of magnetohydrodynamics is given. The spectral stability of the traveling waves generated in the saddle–node bifurcation is studied via an Evans function approach. It is shown that generically one real eigenvalue of the linearization of the viscous conservation law around the parametrized family of traveling waves changes its sign at the bifurcation point. Hence this bifurcation describes the basic mechanism of a stable traveling wave which becomes unstable in a saddle–node bifurcation.
Highlights ► A saddle–node bifurcation of shock waves in viscous conservation laws is studied. ► The saddle–node bifurcation is described through Melnikov integrals. ► The stability of the bifurcating solutions is studied by the Evans function method. ► The Evans function is analyzed by using geometric singular perturbation theory. ► New connections between derivatives of Evans and Melnikov functions are established.

Abstract: Publication date: 15 November 2012
Source:Physica D: Nonlinear Phenomena, Volume 241, Issue 22
Author(s): S. Coombes , R. Thul , K.C.A. Wedgwood
Large scale studies of spiking neural networks are a key part of modern approaches to understanding the dynamics of biological neural tissue. One approach in computational neuroscience has been to consider the detailed electrophysiological properties of neurons and build vast computational compartmental models. An alternative has been to develop minimal models of spiking neurons with a reduction in the dimensionality of both parameter and variable space that facilitates more effective simulation studies. In this latter case the single neuron model of choice is often a variant of the classic integrate-and-fire model, which is described by a nonsmooth dynamical system. In this paper we review some of the more popular spiking models of this class and describe the types of spiking pattern that they can generate (ranging from tonic to burst firing). We show that a number of techniques originally developed for the study of impact oscillators are directly relevant to their analysis, particularly those for treating grazing bifurcations. Importantly we highlight one particular single neuron model, capable of generating realistic spike trains, that is both computationally cheap and analytically tractable. This is a planar nonlinear integrate-and-fire model with a piecewise linear vector field and a state dependent reset upon spiking. We call this the PWL-IF model and analyse it at both the single neuron and network level. The techniques and terminology of nonsmooth dynamical systems are used to flesh out the bifurcation structure of the single neuron model, as well as to develop the notion of Lyapunov exponents. We also show how to construct the phase response curve for this system, emphasising that techniques in mathematical neuroscience may also translate back to the field of nonsmooth dynamical systems. The stability of periodic spiking orbits is assessed using a linear stability analysis of spiking times. At the network level we consider linear coupling between voltage variables, as would occur in neurobiological networks with gap-junction coupling, and show how to analyse the properties (existence and stability) of both the asynchronous and synchronous states. In the former case we use a phase-density technique that is valid for any large system of globally coupled limit cycle oscillators, whilst in the latter we develop a novel technique that can handle the nonsmooth reset of the model upon spiking. Finally we discuss other aspects of neuroscience modelling that may benefit from further translation of ideas from the growing body of knowledge on nonsmooth dynamics.
Highlights ► A new piecewise linear model of a spiking neuron. ► Analysis of single neuron dynamics. ► Analysis of linearly coupled network.

Abstract: Publication date: 1 December 2012
Source:Physica D: Nonlinear Phenomena, Volume 241, Issues 23–24
Author(s): Leonid Chekhov , Marta Mazzocco
In this paper, we study the Goldman bracket between geodesic length functions both on a Riemann surface Σ g , s , 0 of genus g with s = 1 , 2 holes and on a Riemann sphere Σ 0 , 1 , n with one hole and n orbifold points of order two. We show that the corresponding Teichmüller spaces T g , s , 0 and T 0 , 1 , n are realised as real slices of degenerated symplectic leaves in the Dubrovin–Ugaglia Poisson algebra of upper-triangular matrices S with 1 on the diagonal.
Highlights ► We study certain symplectic leaves in the Poisson algebra of Stokes matrices. ► This Poisson algebra is given by the Dubrovin–Ugaglia bracket. ► This coincides with the Goldman bracket in the Teichmüller theory. ► We prove that the Teichmüller spaces belong to degenerated symplectic leaves. ► Their complex dimension is equal to the real dimension of the Teichmüller space.

Abstract: Publication date: 1 August 2013
Source:Physica D: Nonlinear Phenomena, Volumes 256–257
Author(s): Erik Plahte , Arne B. Gjuvsland , Stig W. Omholt
A future quantitative genetics theory should link genetic variation to phenotypic variation in a causally cohesive way based on how genes actually work and interact. We provide a theoretical framework for predicting and understanding the manifestation of genetic variation in haploid and diploid regulatory networks with arbitrary feedback structures and intra-locus and inter-locus functional dependencies. Using results from network and graph theory, we define propagation functions describing how genetic variation in a locus is propagated through the network, and show how their derivatives are related to the network’s feedback structure. Similarly, feedback functions describe the effect of genotypic variation of a locus on itself, either directly or mediated by the network. A simple sign rule relates the sign of the derivative of the feedback function of any locus to the feedback loops involving that particular locus. We show that the sign of the phenotypically manifested interaction between alleles at a diploid locus is equal to the sign of the dominant feedback loop involving that particular locus, in accordance with recent results for a single locus system. Our results provide tools by which one can use observable equilibrium concentrations of gene products to disclose structural properties of the network architecture. Our work is a step towards a theory capable of explaining the pleiotropy and epistasis features of genetic variation in complex regulatory networks as functions of regulatory anatomy and functional location of the genetic variation.

Abstract: Publication date: 1 September 2013
Source:Physica D: Nonlinear Phenomena, Volume 258
Author(s): Mohamed H.M. Sulman , Helga S. Huntley , B.L. Lipphardt Jr. , A.D. Kirwan Jr.
Finite-time Lyapunov exponents (FTLE) are often used to identify Lagrangian Coherent Structures (LCS). Most applications are confined to flows on two-dimensional (2D) surfaces where the LCS are characterized as curves. The extension to three-dimensional (3D) flows, whose LCS are 2D structures embedded in a 3D volume, is theoretically straightforward. However, in geophysical flows at regional scales, full prognostic computation of the evolving 3D velocity field is not computationally feasible. The vertical or diabatic velocity, then, is either ignored or estimated as a diagnostic quantity with questionable accuracy. Even in cases with reliable 3D velocities, it may prove advantageous to minimize the computational burden by calculating trajectories from velocities on carefully chosen surfaces only. When reliable 3D velocity information is unavailable or one velocity component is explicitly ignored, a reduced FTLE form to approximate 2D LCS surfaces in a 3D volume is necessary. The accuracy of two reduced FTLE formulations is assessed here using the ABC flow and a 3D quadrupole flow as test models. One is the standard approach of knitting together FTLE patterns obtained on adjacent surfaces. The other is a new approximation accounting for the dispersion due to vertical ( u , v ) shear. The results are compared with those obtained from the full 3D velocity field. We introduce two diagnostic quantities to identify situations when a fully 3D computation is required for an accurate determination of the 2D LCS. For the ABC flow, we found the full 3D calculation to be necessary unless the vertical ( u , v ) shear is sufficiently small. However, both methods compare favorably with the 3D calculation for the quadrupole model scaled to typical open ocean conditions.

Abstract: Publication date: 1 October 2013
Source:Physica D: Nonlinear Phenomena, Volume 260
Author(s): Martin Burger , Jan Haškovec , Marie-Therese Wolfram
We introduce two models of biological aggregation, based on randomly moving particles with individual stochasticity depending on the perceived average population density in their neighborhood. In the first-order model the location of each individual is subject to a density-dependent random walk, while in the second-order model the density-dependent random walk acts on the velocity variable, together with a density-dependent damping term. The main novelty of our models is that we do not assume any explicit aggregative force acting on the individuals; instead, aggregation is obtained exclusively by reducing the individual stochasticity in response to higher perceived density. We formally derive the corresponding mean-field limits, leading to nonlocal degenerate diffusions. Then, we carry out the mathematical analysis of the first-order model, in particular, we prove the existence of weak solutions and show that it allows for measure-valued steady states. We also perform linear stability analysis and identify conditions for pattern formation. Moreover, we discuss the role of the nonlocality for well-posedness of the first-order model. Finally, we present results of numerical simulations for both the first- and second-order model on the individual-based and continuum levels of description.

Abstract: Publication date: 15 February 2014
Source:Physica D: Nonlinear Phenomena, Volume 269
Author(s): James Hook
Low pass filters, which are used to remove high frequency noise from time series data, smooth the signals they are applied to. In this paper we examine the action of low pass filters on discontinuous or non-differentiable signals from non-smooth dynamical systems. We show that the application of such a filter is equivalent to a change of variables, which transforms the non-smooth system into a smooth one. We examine this smoothing action on a variety of examples and demonstrate how it is useful in the calculation of a non-smooth system’s Lyapunov spectrum.

Abstract: Publication date: 1 March 2014
Source:Physica D: Nonlinear Phenomena, Volume 270
Author(s): Yoji Kawamura
We formulate a theory for the collective phase description of globally coupled noisy limit-cycle oscillators exhibiting macroscopic rhythms. Collective phase equations describing such macroscopic rhythms are derived by means of a two-step phase reduction. The collective phase sensitivity and collective phase coupling functions, which quantitatively characterize the macroscopic rhythms, are illustrated using three representative models of limit-cycle oscillators. As an important result of the theory, we demonstrate noise-induced anti-phase synchronization between macroscopic rhythms by direct numerical simulations of the three models.

Abstract: Publication date: 15 April 2014
Source:Physica D: Nonlinear Phenomena, Volumes 273–274
Author(s): Mike R. Jeffrey
Sharp switches in behaviour, like impacts, stick–slip motion, or electrical relays, can be modelled by differential equations with discontinuities. A discontinuity approximates fine details of a switching process that lie beyond a bulk empirical model. The theory of piecewise-smooth dynamics describes what happens assuming we can solve the system of equations across its discontinuity. What this typically neglects is that effects which are vanishingly small outside the discontinuity can have an arbitrarily large effect at the discontinuity itself. Here we show that such behaviour can be incorporated within the standard theory through nonlinear terms, and these introduce multiple sliding modes. We show that the nonlinear terms persist in more precise models, for example when the discontinuity is smoothed out. The nonlinear sliding can be eliminated, however, if the model contains an irremovable level of unknown error, which provides a criterion for systems to obey the standard Filippov laws for sliding dynamics at a discontinuity.

Abstract: Publication date: 1 May 2014
Source:Physica D: Nonlinear Phenomena, Volume 275
Author(s): O. Stiller
Most data assimilation (DA) methods define the analysis state (i.e., the optimal state for initializing a numerical model) through a quadratic cost function which penalizes both the differences to a model prior (called background state) and the distance to the observations. This paper studies the impact of observation and background error characteristics on the ability to reconstruct spatially localized features with such methods. While the density of the data employed in the DA process gives an upper limit for the spatial reconstruction, this limit can generally only be achieved if the observations are sufficiently precise. This work discusses how finite observation errors (for given background error statistics) degrade the spatial resolution of the analysis state. For this it expands the cost function minimum into a weighted sum over pseudo inverse (PI) solutions each of which corresponds to a different subset of the available observations (i.e., only a subset of the observations is considered for each of these terms, respectively). Observation errors occur only in the weighting factors of this expansion and therefore determine the extent to which observational information is included in the analysis state. More precisely, the weighting factors of the different PIs can be written in terms of normalized observation errors and the determinant of a correlation matrix which characterizes the overlap of the corresponding observation operators. The presented mathematical results are illustrated with a simple model problem which explicitly shows how the reconstruction of a localized feature depends on observation errors as well as the observation operators’ overlap. The findings of this work generally demonstrate that large observation errors do not only decrease the overall weight which the respective observations obtain in the DA process, they especially reduce the DA systems capability to obtain spatially localized information. Small observation errors are particularly important when processing strongly non-local observations as they are typically obtained from passive remote sensing measurements. These have the potential to smear out signals from localized sources over large regions in model space. Generally, observation errors have to be smaller the more the respective observation operators overlap.

Abstract: Publication date: 15 June 2014
Source:Physica D: Nonlinear Phenomena, Volumes 278–279
Author(s): J. Einarsson , J.R. Angilella , B. Mehlig
The orientational dynamics of weakly inertial axisymmetric particles in a steady flow is investigated. We derive an asymptotic equation of motion for the unit axial vector along the particle symmetry axis, valid for small Stokes number St , and for any axisymmetric particle in any steady linear viscous flow. This reduced dynamics is analysed in two ways, both pertain to the case of a simple shear flow. In this case inertia induces a coupling between precession and nutation. This coupling affects the dynamics of the particle, breaks the degeneracy of the Jeffery orbits, and creates two limiting periodic orbits. We calculate the leading-order Floquet exponents of the limiting periodic orbits and show analytically that prolate objects tend to a tumbling orbit, while oblate objects tend to a log-rolling orbit, in agreement with previous analytical and numerical results. Second, we analyse the role of the limiting orbits when rotational noise is present. We formulate the Fokker–Planck equation describing the orientational distribution of an axisymmetric particle, valid for small St and general Péclet number Pe . Numerical solutions of the Fokker–Planck equation, obtained by means of expansion in spherical harmonics, show that stationary orientational distributions are close to the inertia-free case when Pe St ≪ 1 , whereas they are determined by inertial effects, though small, when Pe ≫ 1 / St ≫ 1 .

Abstract: Publication date: 1 February 2015
Source:Physica D: Nonlinear Phenomena, Volumes 292–293
Author(s): Milton C. Lopes Filho , Helena J. Nussenzveig Lopes , Edriss S. Titi , Aibin Zang
In this article we consider the Euler- α system as a regularization of the incompressible Euler equations in a smooth, two-dimensional, bounded domain. For the limiting Euler system we consider the usual non-penetration boundary condition, while, for the Euler- α regularization, we use velocity vanishing at the boundary. We also assume that the initial velocities for the Euler- α system approximate, in a suitable sense, as the regularization parameter α → 0 , the initial velocity for the limiting Euler system. For small values of α , this situation leads to a boundary layer, which is the main concern of this work. Our main result is that, under appropriate regularity assumptions, and despite the presence of this boundary layer, the solutions of the Euler- α system converge, as α → 0 , to the corresponding solution of the Euler equations, in L 2 in space, uniformly in time. We also present an example involving parallel flows, in order to illustrate the indifference to the boundary layer of the α → 0 limit, which underlies our work.

Abstract: Publication date: Available online 6 September 2014
Source:Physica D: Nonlinear Phenomena
Author(s): M. Tamborrino , L. Sacerdote , M. Jacobsen
We consider the multivariate point process determined by the crossing times of the components of a multivariate jump process through a multivariate boundary, assuming to reset each component to an initial value after its boundary crossing. We prove that this point process converges weakly to the point process determined by the crossing times of the limit process. This holds for both diffusion and deterministic limit processes. The almost sure convergence of the first passage times under the almost sure convergence of the processes is also proved. The particular case of a multivariate Stein process converging to a multivariate Ornstein–Uhlenbeck process is discussed as a guideline for applying diffusion limits for jump processes. We apply our theoretical findings to neural network modeling. The proposed model gives a mathematical foundation to the generalization of the class of Leaky Integrate-and-Fire models for single neural dynamics to the case of a firing network of neurons. This will help future study of dependent spike trains.

Abstract: Publication date: Available online 14 August 2014
Source:Physica D: Nonlinear Phenomena
Author(s): Hisa-Aki Tanaka
A physical limit of entrainability of nonlinear oscillators is considered for an external weak signal (forcing). This limit of entrainability is characterized by the optimization problem maximizing the width of the Arnold tongue (the frequency-locking range vs forcing magnitude) under certain practical constraints. Here we show a solution to this optimization problem, thanks to a direct link to Hölder’s inequality. This solution defines an ideal forcing realizing the entrainment limit, and as the result, a fundamental limit of entrainment is clarified as follows. For 1 : 1 entrainment, we obtain (i) a construction of the global optimal forcing and a condition for its uniqueness in L p -space with p > 1 , and (ii) a construction of the global optimal pulse-like forcings in L 1 -space, and for m : n entrainment ( m ≠ n ), some informations about the non-existence of the ideal forcing. (iii) In addition, we establish definite algorithms for obtaining the global optimal forcings for 1 < p ≤ ∞ and these pulse-like forcings for p = 1 . These theoretical findings are verified by systematic, extensive numerical calculations and simulations.